Invariance of an endo-class under the essentially tame Jacquet-Langlands correspondence (Q2360860)

Let \(F\) be a non-Archimedean local field with finite residue field. Let \(D\) be a central \(F\)-division algebra of dimension \(d^2\), \(d \geq 1\). The Jacquet-Langlands correspondence is a canonical bijection between the set of equivalence classes of essentially square integrable representations of \(\mathrm{GL}_n(F)\) and the set of equivalence classes of essentially square integrable representations of \(G=\mathrm{GL}_m(D)\), where \(n=md\). The notion of an endo-class over \(F\) of a simple character for \(\mathrm{GL}_n(F)\) was introduced by \textit{C. J. Bushnell} and \textit{G. Henniart} in [Publ. Math., Inst. Hautes Étud. Sci. 83, 105--233 (1996; Zbl 0878.11042)]. \textit{P. Broussous} et al. in [Doc. Math., J. DMV 17, 23--77 (2012; Zbl 1280.22018)] generalized it to the inner forms of \(G\). They conjectured that the Jacquet-Langlands correspondence preserves endo-classes. This paper verifies the conjecture in the essentially tame case.